Name
ID
MATH 251
Exam 2
Spring 2006
Sections 506
P. Yasskin
Multiple Choice: (5 points each. No part credit.)
1. Compute
∫∫ xy 2 dx dy
1-9
/45
10
/15
11
/10
12
/20
13
/15
Total
/105
over the region bounded by x = y 2 and x = 4.
a. 256
21
b. 512
21
160
c.
21
272
d.
15
e. 544
15
2.
⃗ = (xz 2 , yz 2 , z 3 ), compute
If F
∫∫∫ ∇⃗ ⋅ F dV
over the solid hemisphere 0 ≤ z ≤
a.
b.
c.
d.
e.
4 − x2 − y2 .
64π
3
64π
15
32π
15
320π
3
640π
3
1
3.
Compute
∫∫
1 dA over the region inside the
y
circle given in polar coordinates by r = 2 sin θ.
y
π
4
π
b.
2
c. π
2
a.
1
d. 2π
e. undefined
-1
0
1
x
4. Find the mass of the solid inside the cylinder x 2 + y 2 = 4 between z = 0 and
z = 3 − x2 + y2
if the density is δ =
x2 + y2 .
a. π
b.
c.
d.
e.
4
π
2
π
2π
8π
5. Find the z-component of the center of mass of the solid inside the cylinder x 2 + y 2 = 4
between z = 0 and z = 3 − x 2 + y 2
if the density is δ =
x2 + y2 .
a. 32π
5
b. 16π
c. 4
5
5
5
32π
e. 5
4
d.
2
6. Compute
2
2
∫0 ∫y ex
2
dx dy
a. 1 (1 − e −2 )
2
b. 1 e −2
2
c. 1 (e 2 − 1)
2
d. 1 e 4
2
e. 1 (e 4 − 1)
2
7. Compute
0 ≤ t ≤ π.
∮ F⃗ ⋅ ds⃗
⃗ = (−y, x) along the curve ⃗r(t) = (sin t cos t, sin 2 t) for
for the vector field F
⃗ ⋅ ⃗v.
HINT: Factor F
a. π
4
b. π
2
c. π
d. 2π
e. 4π
3
⃗ ⋅∇
⃗ f.
8. If f = sin(xyz) compute ∇
a.
b.
c.
d.
e.
9.
0
sin(xyz)(−y 2 z 2 , x 2 z 2 , −x 2 y 2 )
sin(xyz)(−y 2 z 2 , −x 2 z 2 , −x 2 y 2 )
−(y 2 z 2 + x 2 z 2 + x 2 y 2 ) sin(xyz)
−(y 2 z 2 − x 2 z 2 + x 2 y 2 ) sin(xyz)
Find the area of the polar ice cap given in
spherical coordinates by 0 ≤ ϕ ≤ π
6
given that the radius of the earth is 4000 miles.
HINT: Parametrize the piece of the sphere
and find the normal vector.
a. 4000 2 π 2 − 3
b. 4000 2 1 −
3
2
c. 4000 2 2 − 3
d. 4000 2 π 1 −
3
2
e. 4000 2 π
4
Work Out: (Part credit possible. Show all work.)
10.
15 points The average of a function f on a curve C is f ave = 1 ∫ f ds where L is
L C
the length of the curve. Find the average temperature T ave on the curve
⃗r(t) = t, t 2 , 2 t 3
between (0, 0, 0) and 1, 1, 2
if the temperature is T = xy 2 + 3yz.
3
3
11.
10 points Compute ∫∫∫ x dV over the triangular pyramid with
vertices (0, 0, 0), (1, 0, 0), (0, 2, 0), and (0, 0, 4).
5
12.
20 points
Consider the saddle z = xy
⃗ (u, v) = (u, v, uv).
which may be parametrized by R
Compute
∫∫ ∇⃗ × F⃗ ⋅ dS⃗
⃗ = (−yz, xz, 0) over the
for F
piece of the saddle with 0 ≤ x ≤ 2 and 0 ≤ y ≤ 3
with normal pointing up.
HINT: First compute each of the following:
⃗, ∇
⃗ R
⃗ (u, v) , ⃗e u , ⃗e v , N
⃗, ∇
⃗⋅N
⃗
⃗ ×F
⃗ ×F
⃗ ×F
∇
6
13.
15 points
Compute
∫∫
y
x dx dy over the diamond shaped region R bounded by
R
y = 1x ,
y = 4x ,
y = x,
y = 9x
y
FULL CREDIT for integrating in the curvilinear
coordinates (u, v) where
y
u 2 = xy and v 2 = x .
(Solve for x and y.)
HALF CREDIT for integrating in rectangular coordinates.
x
7